Abstract

We introduce the sequence given by generating function and establish some explicit formulas for the sequence . Several identities involving the sequence , Stirling numbers, Euler polynomials, and the central factorial numbers are also presented.

1. Introduction and Definitions

For a real or complex parameter , the generalized Euler polynomials are defined by the following generating function (see [1–4])
Obviously, we have
in terms of the classical Euler polynomials , being the set of positive integers. The classical Euler numbers are given by the following:

The so-called the generalized Euler numbers are defined by (see [3, 5])
In fact, are the Euler numbers of order , being the set of integers. The numbers are the ordinary Euler numbers.

Zhi-Hong Sun introduces the sequence similar to Euler numbers as follows (see [6, 7]):
where (and in what follows) is the greatest integer not exceeding .

Clearly, for . The first few values of are shown below

The sequence is related to the classical Bernoulli polynomials (see [8–11]) and the classical Euler polynomials . Zhi-Hong Sun gets the generating function of and deduces many identities involving . As example, (see [6]),

Similarly, we can define the generalized sequence . For a real or complex parameter , the generalized sequence is defined by the following generating function:
Obviously,
By using (10), we can obtain

We now return to the Stirling numbers of the first kind, which are usually defined by (see [2, 5, 8, 11, 12])
or by the following generating function:

3. Proofs of Theorems

Proof of Theorem 1. By (10), (13), and (18), we have
which readily yields
This completes the proof of Theorem 1.

Proof of Theorem 5. By (10), we have
and , thus
By Theorem 1 and comparing the coefficient of on both sides of (35), we get
Again, by taking in Theorem 1, we have
By (36) and (37), we immediately obtain (27). This completes the proof of Theorem 5.

Proof of Theorem 6. By applying Theorem 1, we have
On the other hand, it follows from (10) that
By using (38) and (39), we find that
We now note that
Hence,
yields
Comparing the coefficient of on both sides of (43), we immediately get (28). This completes the proof of Theorem 6.

Proof of Theorem 7. Consider
Thus,
By (42) and (45) we obtain (29). This completes the proof of Theorem 7.

Proof of Theorem 8. By using (7), we have
Thus
That is,
Comparing the coefficient of on both sides of (48), we get the following:
By (49) we immediately obtain (30). This completes the proof of Theorem 8.

Proof of Theorem 9. By integrating (7) with respect to from 0 to 1, we have
By (50) and ( is constant), we have (31). This completes the proof of Theorem 9.

Acknowledgments

This work is partly supported by the Social Science Foundation (no. 2012YB03) of Huizhou University and the Key Discipline Foundation (no. JG2011019) of Huizhou University.